Actual source code: ex26.c
2: static char help[] = "Transient nonlinear driven cavity in 2d.\n\
3: \n\
4: The 2D driven cavity problem is solved in a velocity-vorticity formulation.\n\
5: The flow can be driven with the lid or with bouyancy or both:\n\
6: -lidvelocity <lid>, where <lid> = dimensionless velocity of lid\n\
7: -grashof <gr>, where <gr> = dimensionless temperature gradent\n\
8: -prandtl <pr>, where <pr> = dimensionless thermal/momentum diffusity ratio\n\
9: -contours : draw contour plots of solution\n\n";
10: /*
11: See src/snes/tutorials/ex19.c for the steady-state version.
13: There used to be a SNES example (src/snes/tutorials/ex27.c) that
14: implemented this algorithm without using TS and was used for the numerical
15: results in the paper
17: Todd S. Coffey and C. T. Kelley and David E. Keyes, Pseudotransient
18: Continuation and Differential-Algebraic Equations, 2003.
20: That example was removed because it used obsolete interfaces, but the
21: algorithms from the paper can be reproduced using this example.
23: Note: The paper describes the algorithm as being linearly implicit but the
24: numerical results were created using nonlinearly implicit Euler. The
25: algorithm as described (linearly implicit) is more efficient and is the
26: default when using TSPSEUDO. If you want to reproduce the numerical
27: results from the paper, you'll have to change the SNES to converge the
28: nonlinear solve (e.g., -snes_type newtonls). The DAE versus ODE variants
29: are controlled using the -parabolic option.
31: Comment preserved from snes/tutorials/ex27.c, since removed:
33: [H]owever Figure 3.1 was generated with a slightly different algorithm
34: (see targets runex27 and runex27_p) than described in the paper. In
35: particular, the described algorithm is linearly implicit, advancing to
36: the next step after one Newton step, so that the steady state residual
37: is always used, but the figure was generated by converging each step to
38: a relative tolerance of 1.e-3. On the example problem, setting
39: -snes_type ksponly has only minor impact on number of steps, but
40: significantly reduces the required number of linear solves.
42: See also https://lists.mcs.anl.gov/pipermail/petsc-dev/2010-March/002362.html
43: */
45: /* ------------------------------------------------------------------------
47: We thank David E. Keyes for contributing the driven cavity discretization
48: within this example code.
50: This problem is modeled by the partial differential equation system
52: - Lap(U) - Grad_y(Omega) = 0
53: - Lap(V) + Grad_x(Omega) = 0
54: Omega_t - Lap(Omega) + Div([U*Omega,V*Omega]) - GR*Grad_x(T) = 0
55: T_t - Lap(T) + PR*Div([U*T,V*T]) = 0
57: in the unit square, which is uniformly discretized in each of x and
58: y in this simple encoding.
60: No-slip, rigid-wall Dirichlet conditions are used for [U,V].
61: Dirichlet conditions are used for Omega, based on the definition of
62: vorticity: Omega = - Grad_y(U) + Grad_x(V), where along each
63: constant coordinate boundary, the tangential derivative is zero.
64: Dirichlet conditions are used for T on the left and right walls,
65: and insulation homogeneous Neumann conditions are used for T on
66: the top and bottom walls.
68: A finite difference approximation with the usual 5-point stencil
69: is used to discretize the boundary value problem to obtain a
70: nonlinear system of equations. Upwinding is used for the divergence
71: (convective) terms and central for the gradient (source) terms.
73: The Jacobian can be either
74: * formed via finite differencing using coloring (the default), or
75: * applied matrix-free via the option -snes_mf
76: (for larger grid problems this variant may not converge
77: without a preconditioner due to ill-conditioning).
79: ------------------------------------------------------------------------- */
81: /*
82: Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
83: Include "petscts.h" so that we can use TS solvers. Note that this
84: file automatically includes:
85: petscsys.h - base PETSc routines petscvec.h - vectors
86: petscmat.h - matrices
87: petscis.h - index sets petscksp.h - Krylov subspace methods
88: petscviewer.h - viewers petscpc.h - preconditioners
89: petscksp.h - linear solvers petscsnes.h - nonlinear solvers
90: */
91: #include <petscts.h>
92: #include <petscdm.h>
93: #include <petscdmda.h>
95: /*
96: User-defined routines and data structures
97: */
98: typedef struct {
99: PetscScalar u,v,omega,temp;
100: } Field;
102: PetscErrorCode FormIFunctionLocal(DMDALocalInfo*,PetscReal,Field**,Field**,Field**,void*);
104: typedef struct {
105: PetscReal lidvelocity,prandtl,grashof; /* physical parameters */
106: PetscBool parabolic; /* allow a transient term corresponding roughly to artificial compressibility */
107: PetscReal cfl_initial; /* CFL for first time step */
108: } AppCtx;
110: PetscErrorCode FormInitialSolution(TS,Vec,AppCtx*);
112: int main(int argc,char **argv)
113: {
114: AppCtx user; /* user-defined work context */
115: PetscInt mx,my,steps;
116: PetscErrorCode ierr;
117: TS ts;
118: DM da;
119: Vec X;
120: PetscReal ftime;
121: TSConvergedReason reason;
123: PetscInitialize(&argc,&argv,(char*)0,help);
124: TSCreate(PETSC_COMM_WORLD,&ts);
125: DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,4,4,PETSC_DECIDE,PETSC_DECIDE,4,1,0,0,&da);
126: DMSetFromOptions(da);
127: DMSetUp(da);
128: TSSetDM(ts,(DM)da);
130: DMDAGetInfo(da,0,&mx,&my,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,
131: PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);
132: /*
133: Problem parameters (velocity of lid, prandtl, and grashof numbers)
134: */
135: user.lidvelocity = 1.0/(mx*my);
136: user.prandtl = 1.0;
137: user.grashof = 1.0;
138: user.parabolic = PETSC_FALSE;
139: user.cfl_initial = 50.;
141: PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Driven cavity/natural convection options","");
142: PetscOptionsReal("-lidvelocity","Lid velocity, related to Reynolds number","",user.lidvelocity,&user.lidvelocity,NULL);
143: PetscOptionsReal("-prandtl","Ratio of viscous to thermal diffusivity","",user.prandtl,&user.prandtl,NULL);
144: PetscOptionsReal("-grashof","Ratio of bouyant to viscous forces","",user.grashof,&user.grashof,NULL);
145: PetscOptionsBool("-parabolic","Relax incompressibility to make the system parabolic instead of differential-algebraic","",user.parabolic,&user.parabolic,NULL);
146: PetscOptionsReal("-cfl_initial","Advective CFL for the first time step","",user.cfl_initial,&user.cfl_initial,NULL);
147: PetscOptionsEnd();
149: DMDASetFieldName(da,0,"x-velocity");
150: DMDASetFieldName(da,1,"y-velocity");
151: DMDASetFieldName(da,2,"Omega");
152: DMDASetFieldName(da,3,"temperature");
154: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
155: Create user context, set problem data, create vector data structures.
156: Also, compute the initial guess.
157: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
159: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
160: Create time integration context
161: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
162: DMSetApplicationContext(da,&user);
163: DMDATSSetIFunctionLocal(da,INSERT_VALUES,(DMDATSIFunctionLocal)FormIFunctionLocal,&user);
164: TSSetMaxSteps(ts,10000);
165: TSSetMaxTime(ts,1e12);
166: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
167: TSSetTimeStep(ts,user.cfl_initial/(user.lidvelocity*mx));
168: TSSetFromOptions(ts);
170: PetscPrintf(PETSC_COMM_WORLD,"%Dx%D grid, lid velocity = %g, prandtl # = %g, grashof # = %g\n",mx,my,(double)user.lidvelocity,(double)user.prandtl,(double)user.grashof);
172: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
173: Solve the nonlinear system
174: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
176: DMCreateGlobalVector(da,&X);
177: FormInitialSolution(ts,X,&user);
179: TSSolve(ts,X);
180: TSGetSolveTime(ts,&ftime);
181: TSGetStepNumber(ts,&steps);
182: TSGetConvergedReason(ts,&reason);
184: PetscPrintf(PETSC_COMM_WORLD,"%s at time %g after %D steps\n",TSConvergedReasons[reason],(double)ftime,steps);
186: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
187: Free work space. All PETSc objects should be destroyed when they
188: are no longer needed.
189: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
190: VecDestroy(&X);
191: DMDestroy(&da);
192: TSDestroy(&ts);
194: PetscFinalize();
195: return 0;
196: }
198: /* ------------------------------------------------------------------- */
200: /*
201: FormInitialSolution - Forms initial approximation.
203: Input Parameters:
204: user - user-defined application context
205: X - vector
207: Output Parameter:
208: X - vector
209: */
210: PetscErrorCode FormInitialSolution(TS ts,Vec X,AppCtx *user)
211: {
212: DM da;
213: PetscInt i,j,mx,xs,ys,xm,ym;
214: PetscReal grashof,dx;
215: Field **x;
217: grashof = user->grashof;
218: TSGetDM(ts,&da);
219: DMDAGetInfo(da,0,&mx,0,0,0,0,0,0,0,0,0,0,0);
220: dx = 1.0/(mx-1);
222: /*
223: Get local grid boundaries (for 2-dimensional DMDA):
224: xs, ys - starting grid indices (no ghost points)
225: xm, ym - widths of local grid (no ghost points)
226: */
227: DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);
229: /*
230: Get a pointer to vector data.
231: - For default PETSc vectors, VecGetArray() returns a pointer to
232: the data array. Otherwise, the routine is implementation dependent.
233: - You MUST call VecRestoreArray() when you no longer need access to
234: the array.
235: */
236: DMDAVecGetArray(da,X,&x);
238: /*
239: Compute initial guess over the locally owned part of the grid
240: Initial condition is motionless fluid and equilibrium temperature
241: */
242: for (j=ys; j<ys+ym; j++) {
243: for (i=xs; i<xs+xm; i++) {
244: x[j][i].u = 0.0;
245: x[j][i].v = 0.0;
246: x[j][i].omega = 0.0;
247: x[j][i].temp = (grashof>0)*i*dx;
248: }
249: }
251: /*
252: Restore vector
253: */
254: DMDAVecRestoreArray(da,X,&x);
255: return 0;
256: }
258: PetscErrorCode FormIFunctionLocal(DMDALocalInfo *info,PetscReal ptime,Field **x,Field **xdot,Field **f,void *ptr)
259: {
260: AppCtx *user = (AppCtx*)ptr;
261: PetscInt xints,xinte,yints,yinte,i,j;
262: PetscReal hx,hy,dhx,dhy,hxdhy,hydhx;
263: PetscReal grashof,prandtl,lid;
264: PetscScalar u,udot,uxx,uyy,vx,vy,avx,avy,vxp,vxm,vyp,vym;
267: grashof = user->grashof;
268: prandtl = user->prandtl;
269: lid = user->lidvelocity;
271: /*
272: Define mesh intervals ratios for uniform grid.
274: Note: FD formulae below are normalized by multiplying through by
275: local volume element (i.e. hx*hy) to obtain coefficients O(1) in two dimensions.
277: */
278: dhx = (PetscReal)(info->mx-1); dhy = (PetscReal)(info->my-1);
279: hx = 1.0/dhx; hy = 1.0/dhy;
280: hxdhy = hx*dhy; hydhx = hy*dhx;
282: xints = info->xs; xinte = info->xs+info->xm; yints = info->ys; yinte = info->ys+info->ym;
284: /* Test whether we are on the bottom edge of the global array */
285: if (yints == 0) {
286: j = 0;
287: yints = yints + 1;
288: /* bottom edge */
289: for (i=info->xs; i<info->xs+info->xm; i++) {
290: f[j][i].u = x[j][i].u;
291: f[j][i].v = x[j][i].v;
292: f[j][i].omega = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy;
293: f[j][i].temp = x[j][i].temp-x[j+1][i].temp;
294: }
295: }
297: /* Test whether we are on the top edge of the global array */
298: if (yinte == info->my) {
299: j = info->my - 1;
300: yinte = yinte - 1;
301: /* top edge */
302: for (i=info->xs; i<info->xs+info->xm; i++) {
303: f[j][i].u = x[j][i].u - lid;
304: f[j][i].v = x[j][i].v;
305: f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy;
306: f[j][i].temp = x[j][i].temp-x[j-1][i].temp;
307: }
308: }
310: /* Test whether we are on the left edge of the global array */
311: if (xints == 0) {
312: i = 0;
313: xints = xints + 1;
314: /* left edge */
315: for (j=info->ys; j<info->ys+info->ym; j++) {
316: f[j][i].u = x[j][i].u;
317: f[j][i].v = x[j][i].v;
318: f[j][i].omega = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx;
319: f[j][i].temp = x[j][i].temp;
320: }
321: }
323: /* Test whether we are on the right edge of the global array */
324: if (xinte == info->mx) {
325: i = info->mx - 1;
326: xinte = xinte - 1;
327: /* right edge */
328: for (j=info->ys; j<info->ys+info->ym; j++) {
329: f[j][i].u = x[j][i].u;
330: f[j][i].v = x[j][i].v;
331: f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx;
332: f[j][i].temp = x[j][i].temp - (PetscReal)(grashof>0);
333: }
334: }
336: /* Compute over the interior points */
337: for (j=yints; j<yinte; j++) {
338: for (i=xints; i<xinte; i++) {
340: /*
341: convective coefficients for upwinding
342: */
343: vx = x[j][i].u; avx = PetscAbsScalar(vx);
344: vxp = .5*(vx+avx); vxm = .5*(vx-avx);
345: vy = x[j][i].v; avy = PetscAbsScalar(vy);
346: vyp = .5*(vy+avy); vym = .5*(vy-avy);
348: /* U velocity */
349: u = x[j][i].u;
350: udot = user->parabolic ? xdot[j][i].u : 0.;
351: uxx = (2.0*u - x[j][i-1].u - x[j][i+1].u)*hydhx;
352: uyy = (2.0*u - x[j-1][i].u - x[j+1][i].u)*hxdhy;
353: f[j][i].u = udot + uxx + uyy - .5*(x[j+1][i].omega-x[j-1][i].omega)*hx;
355: /* V velocity */
356: u = x[j][i].v;
357: udot = user->parabolic ? xdot[j][i].v : 0.;
358: uxx = (2.0*u - x[j][i-1].v - x[j][i+1].v)*hydhx;
359: uyy = (2.0*u - x[j-1][i].v - x[j+1][i].v)*hxdhy;
360: f[j][i].v = udot + uxx + uyy + .5*(x[j][i+1].omega-x[j][i-1].omega)*hy;
362: /* Omega */
363: u = x[j][i].omega;
364: uxx = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx;
365: uyy = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy;
366: f[j][i].omega = (xdot[j][i].omega + uxx + uyy
367: + (vxp*(u - x[j][i-1].omega)
368: + vxm*(x[j][i+1].omega - u)) * hy
369: + (vyp*(u - x[j-1][i].omega)
370: + vym*(x[j+1][i].omega - u)) * hx
371: - .5 * grashof * (x[j][i+1].temp - x[j][i-1].temp) * hy);
373: /* Temperature */
374: u = x[j][i].temp;
375: uxx = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx;
376: uyy = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy;
377: f[j][i].temp = (xdot[j][i].temp + uxx + uyy
378: + prandtl * ((vxp*(u - x[j][i-1].temp)
379: + vxm*(x[j][i+1].temp - u)) * hy
380: + (vyp*(u - x[j-1][i].temp)
381: + vym*(x[j+1][i].temp - u)) * hx));
382: }
383: }
385: /*
386: Flop count (multiply-adds are counted as 2 operations)
387: */
388: PetscLogFlops(84.0*info->ym*info->xm);
389: return 0;
390: }
392: /*TEST
394: test:
395: args: -da_grid_x 20 -da_grid_y 20 -lidvelocity 100 -grashof 1e3 -ts_max_steps 100 -ts_rtol 1e-3 -ts_atol 1e-3 -ts_type rosw -ts_rosw_type ra3pw -ts_monitor -ts_monitor_solution_vtk 'foo-%03D.vts'
396: requires: !complex !single
398: test:
399: suffix: 2
400: nsize: 4
401: args: -da_grid_x 20 -da_grid_y 20 -lidvelocity 100 -grashof 1e3 -ts_max_steps 100 -ts_rtol 1e-3 -ts_atol 1e-3 -ts_type rosw -ts_rosw_type ra3pw -ts_monitor -ts_monitor_solution_vtk 'foo-%03D.vts'
402: requires: !complex !single
404: test:
405: suffix: 3
406: nsize: 4
407: args: -da_refine 2 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3 -pc_type none -ts_type beuler -ts_monitor -snes_monitor_short -snes_type aspin -da_overlap 4
408: requires: !complex !single
410: test:
411: suffix: 4
412: nsize: 2
413: args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
414: requires: !complex !single
416: test:
417: suffix: asm
418: nsize: 4
419: args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3
420: requires: !complex !single
422: TEST*/